We implement an extension of the pseudofermion functional renormalization group (PFFRG) method for quantum spin systems that takes into account two-loop diagrammatic contributions. An efficient numerical treatment of the additional terms is achieved within a nested graph construction which recombines different one-loop interaction channels. In order to be fully self consistent with respect to self-energy corrections we also include certain three-loop terms of Katanin type. We first apply this formalism to the antiferromagnetic $J_1$-$J_2$ Heisenberg model on the square lattice and benchmark our results against the previous one-loop plus Katanin approach. Even though the RG equations undergo significant modifications when including the two-loop terms, the magnetic phase diagram -- comprising N\'eel ordered and collinear ordered phases separated by a magnetically disordered regime -- remains remarkably unchanged. Only the boundary position between the disordered and the collinear phases is found to be moderately affected by two-loop terms. On the other hand, critical RG scales, which we associate with critical temperatures $T_\text{c}$, are reduced by a factor of $\sim2$ indicating that the two-loop diagrams play a significant role in enforcing the Mermin-Wagner theorem. Improved estimates for critical temperatures are also obtained for the Heisenberg ferromagnet on the 3D simple cubic lattice where errors in $T_\text{c}$ are reduced by $\sim34\%$. These findings have important implications for the quantum phase diagrams calculated within the previous one-loop plus Katanin approach which turn out to be already well-converged.

中文翻译：

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二环贡献在量子自旋系统的伪费米子泛函重整化群方法中的作用

我们为量子自旋系统实现了伪费米子功能重整化组（PFFRG）方法的扩展，该方法考虑了两个环的图解贡献。在重新组合不同的单环交互通道的嵌套图构造中，可以实现对附加项的有效数值处理。为了在自我能量校正方面完全自洽，我们还包括了Katanin类型的某些三环项。我们首先将此形式主义应用于方格上的反铁磁$ J_1 $-$ J_2 $ Heisenberg模型，然后将结果与以前的单环加Katanin方法进行基准比较。即使RG方程在包含两个环项时进行了重大修改，但磁相图-包含N \' 鳗鱼有序相和共线有序相被磁无序状态隔开-保持显着不变。发现只有无序相和共线相之间的边界位置受两环项的影响中等。另一方面，我们将与临界温度$ T_ \ text {c} $关联的临界RG比例降低了$ \ sim2 $，这表明两个回路图在执行Mermin-瓦格纳定理。还获得了3D简单立方晶格上的Heisenberg铁磁体的临界温度的改进估计值，其中$ T_ \ text {c} $中的误差减少了$ \ sim34 \％$。这些发现对先前的单环加Katanin方法计算出的量子相图具有重要意义，事实证明该相图已经很好地收敛了。